
Class. 
Book.. 



T~ A /. ^. I 



Copiglit^?- 



CDEffilGHT DEPOSm 



CONCRETE COMPUTATION 



CHARTS 



BY 



,y^ 



RICHARD Ti DANA 

Mem. Am. Soc« C. £. 
Mem, A. I, M. & M. E, 
Mem, Yale Eng, Assn. 
Ch, Eng,, -Construction Service Company 

AND 

JAMES M. KINGSLEY, B. S. 




NEV\r YORK 
1922 



.^ 



\ 






Copyri^t, 1922, 

by 

R, T. DAM 



^PR 29 1922 
©7A661493 



J 




^ 



^"^ \ 



to 



FOREWORD 

1. The purpose of this Book of Charts is to fur- 
nish to the Designing Engineer^ the Draftsman, the Es- 
timator, and the Student a concise and complete appara- 
tus for Proportioning the Parts and Estimating the Cost 
of Concrete Structures, 

2. The formulas for Moments of Resistance were 
derived in 1904 and are similar in effect to those sug- 
gested by the American Society of Civil Engineers in 
1917. They are here given algebraically and are plotted 
on charts for convenience in computation. In this form 
they can be used to determine the sizes, shapes, compo- 
sition and cost of Concrete beams, slabs, columns, etc, 
with the miniraura expenditure of time and risk of error. 

3. Careful directions accompany these charts, by 
the aid of which the necessary quantities may be read 
directly, thus avoiding the necessity of arithmetical or 
algebraic calculation, minimizing the risks of error al- 
ways present in such calculations, and greatly expediting 
the process of design. The use of these charts does not 
require a knov/ledge of the rather complex mechanical the- 
ory of flexure in Reinforced Concrete Beams, 



(1) 



TABLE OF CONTENTS 



Example : Floor System 

Beam Design 
Descripticin: Charts Cr - 



Chart Cr - 
Charts Cr - 
Chart Cr - 



1 to 4 
5 

6 to 10 
11 

Charts Cr - 12 to 17 
Notes on theory, method of computation and application to field work 
Classification of Concrete for Various Uses 
Tests for Voids 

Description: Charts Cr - 18 to 19 
Example: Cost of 1 cu« yd. of 1:3:6 concrete 
Formulas for reinforced concrete construction 
Standard Notation 
Rectangular "beams 
T-beams 

Beams reinforced for compression 
Shear, bond and web reinforcement 
Columns 
Formulas 

Rectangular beams 
Columns 



PAGE 

4 

5 

6 

7 

8 

9 

10 

10 

11 

12 

13 

14 

14 

14 

14 

14 

15 

15 

15 

15 

15 

15 



CHARTS 



Charts for Bending Moments by M = ^^^ » Wl£ > WL2 

12 10 8 
Charts for Effective Depths of R. C. Beams and Slabs (Logarithmic) 
Chart for Area of Steel Required for Beams and Slabs 
Chart for Effective Depths of R. C. Beams and Slabs (Rectilinear) 
Chart for Effective Depths and Area of Cinder Concrete Slabs 
Chart for Area and Weight of Round Steel Rods 
Chart for Spacing of Standard Round Rods 
Chart for Area and Weight of Square Steel Bars 
Chart for Spacing of Standard Square Bars 
Chart for Round Columns with Longitudinal Reinforcement 
Chart for Rorind Columns with Longitudinal Reinforcement and Hoops 

or Spirals 
Chart for Square Columns with Longitudinal Reinforcement 
Charts for Variables Required in Slab and Beam Formula 

(1) Values of j depending on m and n 

(2) Values of k depending on n and p 

(1) Values of m depending on n and p 

(2) Values of k depending on m and n 

Charts for Proportions of Sand and Stone per Unit Volume of Cement 
E - (Excess of cement and mortar ) being 10 and 15^ 
E - (Excess of cement and mortar ) being 20 and 25^ 
Chart for Amount of Cement Required for Mix as Determined by Cr - 15 

and Cr - 16 
Chart for Compressive Strengths of Concrete of Different Materials 

and Mixtures 
Chart for Cost per Cu. Yd. of Materials for Concrete of Various Mixes 
Chart Showing Building Code Requirements of 26 Cities 
Chart for Weights per Sq, Ft. for Various Thicknesses of Slabs 



Cr 


- 


1,2,3 


Cr 


- 


4 


Cr 


- 


5 


Cr 


- 


6 


Cr 


- 


7 


Cr 


- 


8 


Cr 


- 


8a 


Cr 


•> 


9 


Cr 


- 


9a 


Cr 


- 


10 


Cr 


_ 


11 


Cr 


- 


12 


Cr 


^ 


13 


Cr 


- 


13 


Cr 


> 


14 


Gr 


- 


14 


Cr 


_ 


15 


Cr 


•• 


16 



Cr - 17 



Cr - 


18 


Cr - 


19 


Cr - 


20 


Cr - 


21 



(2) 



EXAMPLE : Design floor system for room 25 x 47 ft, with live load 
of 80 lbs. per sq. ft. 

By using 7 floor beams the area will be divided into 8 slabs of 
6 ft. span. 

Loading L. L. 80# 

D. L, Floor 6C^ (assumed) 

Hanging Ceiling 15# 

Total 15^ 

Moment Enter chart Cr - 1 at bottom at point indicating 155#. 
Run up vertically to the intersection with the inclined line indicating 
span of 6 ft. and from this intersection run horizontally to the right 
or left hand margin where the moment is read as 5550 in. lbs. (Actual- 
ly 5580) 

Slab The moment per ft, of width is 5550 in. lbs. The moment 
per in. of width is 463 in. lbs. (5550-^12) 

d - Enter Cr - 4 at right hand margin at point indicating 463 in. 
lbs. (.463 thousands of in, lbs.). Rixn horizontally to the left to the 
intersection with the inclined line and from this intersection run down 
to the bottom of the sheet where the effective depth is found to tbe 
2.55 in. 

Depth to steel equals 2.55 in. On a thin slab such as this, due to 
difficulty of properly placing steel, etc., it is better practice to make 
the effective depth of the slab soraev,/hat deeper - say 2 3/4 or 3 in. 
With 3/4 in. of concrete belov;^ steel the total thickness of slab will then 
be 3 1/2 or 3 3/4 in. 

As - Enter Cr - 5 at bottom at point indicating effective depth - d 
of 2.55 in. Run up vertically to intersect with inclined line and from 
this intersection nin horizontally to ri^t hand margin where As per in. 
of width is found to be 0.0128 sq. in, or 0.154 per ft. of width. 

Now refer to Cr - 8 and Cr - 9 •niiere the areas of round and square 
rods are shown. If we select 1/4 in. square rods we find the area is 
.0625 sq. in. 

The spacing is found by the formula: 

12 X (Sectional area of bar to be used) 

Spacing a> ^ '- 

^S 
In this case Spacing 12 x ,0625 . „„ , „ . /. . * 

0.154 "■ ^'^^ ^^* *^y * ^/* in? 

Every other bar should be bent up at 45° to run over floor beams 
at upper side of slab. 

* The same result may be determined directly from Cr - 9a, 



(3) 



Beam Design The loading per foot of beam is as follows: 

L. L. 80 X 6~ 480 lbs, 

D. L. Floor (3 3/4 in.) 47 x 6 = 282 

D. L. Ceiling 15 x 6 = 90 
Stem of beam a a 100 (a ssumed) 

Total 952 lbs. 

Moment Enter at 95 at bottom of chart. Run up to inter- 
section Tfith line indicating 26 ft. span and from this intersection 
run horizontally to right hand margin where moment is fotmd to be 
96,000 in. lbs, for loading of 95 lbs. For 950 lbs. the moment would 
be 960,000 in. lbs. (Actually 963300. Error « 0.344?^ which is insig- 
nificant.) 

1. Assume width of stem of beam as 8 in. Then total width of T 
« 8 + 2(3.75) =15.5 in. 

Moment per inch of width equals 960,000 -^15.5 — 6,200 in. lbs. 

From Cr - 4,d =» 9,35. 

2. Assume width of stem of beam as 6 in. Then total Yddth of T 
equals 6 + 2(3.75) -13.5 in. 

Moment per inch of width equals 960,000+13.5 = 7,130 in. lbs. 

From Cr - 4,d ~ 10 in,, which gives a better proportioned and 
more economic section. 

As - From Cr - 5 it is found that for a depth of 10 in., 0.05 sq. in. 
of steel is required per in, of width. Then for the total width of 13.5 
in. 0.675 sq, in, of steel is required 

Use 4 - 7/16 sq. bars Area =^ 4 x .19 =«0.76 sq. in. Two bars are 
to run at bottom of beam throughout and of the other two, one is to be 
bent up at 45° at 1/4 points of span and the other at 45° at points 1/8 
of span from supports. 



(4) 



Cr - 1 is for determining moments by the formula M — 12 * 
This is foruse Tvith the following conditions: 

1, For floor slabs containing over two spans, 

2. For beams of Interior spans. 

In this formula W equals the combined dead and live load per foot; 
L equals the span length in feet. 

In using the diagram, after determining the combined dead and live 
load, enter the diagram at the bottom as indicated by the arrow marked 
1., and follow the vertical line representing the determined load until 
it intersects the inclined line indicating the proper span. Then fol- 
lov^ the horizontal line from this point of intersection over to the 
ri'ght or left margin of the chart where the moment in thousands of inch 
pounds may be read. 



Cr - 2 is for determining moments by the formula M _ ULc • 
This is for use v/ith the following conditions: 10 

1. For end beams and spans. 

2, For beams and slabs v/ith their ends restrained and continuous 
for tvfo spans only. 

In this formula W equals the combined dead and live load per foot; 
L equals the span length in feet. 

In using this diagram proceed according to directions for Cr - 1. 

Cr - 3 is for determining moments by the formula M _. —^ . 
This is for use with the following conditions: 

1, For simple beams and spans. 

In this formula W equals the combined dead and live load per foot; 
L equals the span length in feet. 

In using this diagram proceed according to directions for Cr - 1. 



Cr - 4 is for determining the effective depths of reinforced con- 
crete beams and slabs. 

1. For slabs: Divide the moment in inch pounds as determined 
from Cr - 1, Cr - 2 or Cr - 3 by 12. This will give the moment per 
inch vddth of slab. Having determined this moment, enter the chart on 
the right hand margin where the moments are indicated in thousands of 
inch pounds and, as indicated by the arrow marked 1,, follow the hori- 
zontal line from this point to the intersection of the diagonal line. 



(5) 



From the intersection with this line follov/ the vertical line to the 
bottom niargin where the effective depth (i.e. the depth in inches from 
the top of section to center line of steel) is easily read. 

2, For rectangular beams: After obtaining the moment in inch 
pounds from Cr - 1, Cr - 2 or Cr - 3 the width of beam is assumed and 
the moment is divided by this width thus giving the moment per inch of 
v/idth of beam. After determining this moment enter the chart on the 
right hand margin where the moments are indicated in thousands of inch 
pounds and proceed as in 1 above. 

The width of beam will be largely determined by the lateral spac- 
ing of the parallel reinforcing bars which should not be less than 3 
diameters from center to center. Nor should the distances from the 
side of beam to the center of the nearest bar be less than 2 diameters, 
if two layers of bars are used the clear spacing between the layers of 
bars should be not less than 1 inch. The use of more than tv;o layers 
is not recommended unless the layers are tied together by metal connec- 
tions. As a general rule the width of the beams should not be less than 
1/3 of their depth. 

3, T-beams: In the design of T-beams the width of the stem of the 
beam is assumed and the total width of the beam is taken as the vddth of 
the stem plus twice the thickness of the flange. It should be noted 
that this vadth is considerably less than that allowed by the Joint Com- 
mittee of the American Society of Civil Engineers which recommended that 
the effective width should be determined by the following rules: 

A. Shall not exceed 1/4 of the span length of the beaia. 

B. Its overhanging width on either side of the web shall 
not exceed six times the thickness of the slab. 

By using the lesser width of flange v/e find that lengthy and more or 
less difficult computations to provide against diagonal tension and shear 
are obviated, and it is believed that with the exception of large struc- 
tures the greater simplicity of this method, together with the additional 
security of the more conservative design, justifies the use of the addi- 
tional concrete. 

In determining the depth D of the steel the same steps are taken as 
in the design of slabs and beams as indicated above. 



Cr - 5 is for use in determining the area of steel required for 
slabs and beams. 

In using the diagram enter the chart from the bottom at the point 
indicating the effective depth as determined from Cr - 4 or Cr - 6 and 
follow the vertical line from this point to its intersection with the 
diagonal line. From this intersection follow the horizontal line to 
the right hand margin where the area of steel in square inches is shown. 
This area must be multiplied by the width of the beam and in the case of 
slabs by 12. This will give the total gross sectional area of steel re- 
quired. 

From Cr - 8 and Cr - 9 the proper number of rods to give this area 
may be obtained. 



(6) 



In the case of slabs, Cr - 8a or Cr - 9a will give by inspection 
the proper size, nximber and spacing of rods or bars for required area 
of steel • 



Cr - 6 is for the same purpose as Cr - 4, but it is plotted on rec- 
tilinear paper thus giving a more precise range for slabs and beams of 
depths of more than 6 inches. 



Qr - 7 is for the design of cinder concrete slabs, 

1. On the right hand side it is arranged similarly to Cr - 4 with 
the exception that since cinder concrete as recommended by the Joint Com- 
mittee of the A, S. C. E. should not be used for reinforced concrete 
structures with floor slabs exceeding 8 ft, span, the moments of resist- 
ance are given with a width of 1 ft. It is therefore possible to enter 
the diagram on the right hand margin using the moment in thousands of inch 
poxinds as obtained from Cr - 1, Cr - 2 or Cr - 3 on a vertical line and 
from this point run over on the horizontal line to the inclined line repre- 
senting "effective depth" and from the intersection of this line run down 
on a vertical line to the bottom of the sheet where the depth of slab is 
read, 

II. This portion of the chart is for obtaining the area of steel re- 
quired in cinder concrete slabs. Enter the diagram at the bottom at the 
point indicating the effective depth as determined by I and run up verti- 
cally to the intersection with the inclined line. From this intersection 
run horizontally to the left hand margin of the sheet where the total area 
of steel required per foot of width of slab is read from the scale. 



Cr - 8 is for use in determining the niimber of round steel rods re- 
quired to make up a required area of steel. The diameters of the rods in 
inches are shovm at the bottom of the sheet and by following the operations 
indicated by the arrows on the chart both the weight and area of the rods 
can be obtained. Cr - 8a is self explanatory. 



Cr - 9 is for use in determining the n\jmber of square steel bars re- 
quired to make up a required area of steel. The thickness of the bars in 
inches is shown at the bottom of the sheet and by following the operations 
indicated on the chart both the weight and area of the bars can be obtain- 
ed. Cr - 9a is self explanatory. 



Cr - 10 is for use in designing round coltimns with longitudinal rein- 
forcements to the extent of not less than one per cent and not more than 
four per cent and vfith lateral ties of not less than 1/4 in. in diameter, 
12 inches apart nor more than 16 diameters of the longitudinal bars. The 
chart is used as follows: 

1. Enter the right hand side of the chart using the total load to be 
carried by the column, and, as indicated by the arrow, follovf horizontally 
across to the inclined line indicating the per cent of longitudinal rein- 
forcement- From the intersection of this line, as indicated by arrovj- No. 2, 
follov/ the vertical line to the bottom of the sheet where the effective 
diameter of the nolximn is shown. 



(7) 



JI, Using the effective diameter as obtained above, enter the left 
hand side of the chart at the bottom of the sheet and follow the vertical 
line upwards as indicated by the arrow 1, to the intersection vdth the in- 
clined line indicating th,e same percentage of steel used in obtaining the 
effective diameter. From this intersection follow the horizontal line to 
the left hand margin where the total area of longitudinal steel in square 
inches is shoVirn, 

EXAMPLE ; Design column for load of 200,000 lbs. Diameter limited to 
24 inches. 

Allowing for thickness of 1 1/2 in. for fireproofing, effective dia- 
meter will be limited to 21 inches. 

Entering I at right hand margin at point indicating 200,000 lbs., 
run over to intersection with the vertical line representing diameter of 
21 in. The ratio of steel area to area of concrete is found to be .02* 

Entering II at bottom under 21 rtin up the vertical line to inter- 
section with the inclined line for the .02 ratio of steel area to area of 
concrete, and from this intersection run over horizontally to the left 
hand margin where the area of steel is found to be 6#9 sq. in. 

From Cr - 8 and Cr - 9 the required size and number of rods may be 
figured. 

Using 1 in. sq. rods (Cr - 9) the area of each rod is 1 sq, in. and 
7 rods are required. 



Cr - 11 is for use in designing round colvanns reinforced with not 
less than one per cent nor more than four per cent of longitudinal bars 
and with circular hoops or spirals not less than one per cent of the 
volvime of the concrete contained within the reinforcement. 

In using this diagram the diameter of the column and area of the lon- 
gitudinal steel is obtained in the same manner as in Cr - 10 and in addi- 
tion the cross section of the hoops is obtained by the portion of the dia- 
gram in the lower left hand corner, marked III, as follows: Enter the 
diagram from the bottom using the diameter as determined by I and follow 
the line vertically upward to the intersection with the inclined line in- 
dicating the approximate predetermined spacing of the hoops. From the in- 
tersection with this line follow the horizontal line to the left hand mar- 
gin where the cross sectional area of the hoops is at once read off in 
square inches. 

Select from Cr- 8 or Cr - 9 the size rod nearest to the cross-sec- 
tion found and re-enter the diagram on the left hand margin* The inter- 
section of the horizontal line indicating cross-section of bar selected 
and the vertical line indicating the effective diameter of the col\iran 
gives a point which is used to determine by interpolation the exact spac- 
ing of the hoops. 

In regard to the spacing of the hoops, the Joint Committee of the 
A, S. C. E, recommends that the spacing of the hooping should not be 
more than 1/6 of the enclosed column and preferably not greater than 
1/10 and in no case more than 2 1/2 ±u , 

(8) 



Cr - 12 is for use in designing square columns Vfith longitudinal 
reinforcements to the extent of not less than one per cent and not more 
than four per cent and with lateral ties of not less than 1/4 in# in 
diameter IE tns, apart nor more than 16 diameters of the longitudinal 
"bars. This chart is used as follows: 

I. Enter the right hand side of the chart using the total load 
to be carried "by the column and, as indicated by the arrow, follow hor- 
izontally across to the inclined line indicating the per cent of longi- 
tudinal reinforcement. From the intersection of this line, as indicated 
by arrow No. 2, follow the vertical line to the bottom of the sheet 
where the effective dimensions of the colvimn are shown. 

II. Using the effective dimensions as obtained above, enter the 
left hand side of the chart at the bottom of the sheet and follow the 
vertical line upv/ards as indicated by arrow No. 1 to the intersection 
with the inclined line indicating the same percentage of steel used in 
obtaining the effective dimensions. From this intersection follovv the 
horizontal line to the left hand margin where the total area of longi- 
tudinal steel in square inches is shovm. 



Cr - 13 is for use in determining variables required for utiliz- 
ing the slab and beam formula. The upper part of the chart shows values 
of j as varying for different values of m and n. 

Entering the chart at the bottom with the value of m determined 
for the particular problem, follow the vertical line to its intersection 
v/ith the curved line representing the value of n, and from this inter- 
section follow the horizontal line to the left hand margin "where the val- 
ue of j is shown. 

The lower part of the chart shows values of k for varying combina- 
tions of n and p. The method of using this part of the chart is similar 
to the above. 



Cr - 14 is also for use in determining variables for use in the con- 
crete beam and slab formula. The upper part show^s values of m as depend- 
ing upon different values of n and p. 

The lower part of the chart shows the values of k as depending upon 
m and n. 



Cr - 15 and Cr - 16 are for determining the proportions of sand and 
stone per unit volume of cement for an economic mix. 

Cr - 17 is for detennining the amount of cement required and is for 
use in connection with Cr - 16 and Cr - 15. 

The following notes are abstracted from an article describing the 
theory, method of computation and application of the charts to field prac- 
tice, published in Engineering News, April 20, 1905. 



(9) 



The assxanptions of the theory are as follov/s : 

(1) The voids in the sand should be filled with cement paste. 

(2) The voids in the stone or gravel should he filled vdth mortar. 

(3) There should be a small and definite excess of paste over the 
amo\int necessary to fill the voids in the sand. 

(4) Likevfise there should be a small and definite excess of mortar 
over the amoiint necessary to fill the voids in the stone. 

(5) The voids in the sand and stone are easily determinable. And 
the voids of the stone after being rammed can be determined before addi- 
tion of mortar with sufficient accuracy for all practical purposes, any 
error here being on the safe side. 

(6) The excess of cement serves the two-fold purpose of compensat- 
ing for irregularities in mixing, preventing accidental voids, and of sup- 
plying a coat of cement over the surface of each grain of sand and piece 
of stone. A larg«-excess of cement accomplishes no useful object, makes 
the concrete more liable to cracks and checks, and, in general, reduces 
the density of the mass, besides adding greatly to the cost of the work. 

The extreme stone values given in the diagrams, when the stone voids 
are 35J^ and the cement voids 15%, will rarely be arrived at in practice, 
but the diagrams can be used v/ith perfect confidence and with absolute as- 
surance that the voids will be filled provided that the voids in the sand 
and stone are properly determined in the first instance. It vdll be noted 
that the voids in the stone are supposed to be the same in the first deter- 
minations as after mixing in the mortar. No allowance has been made for 
consolidation by ramming. This does not mean that ramming is unnecessary. 

It is the writer's practice to mix the concrete with enough water to 
make it just quake xinder ramming and then to ram rapidly and briefly. For 
watertight work slightly more water is added. 

Attention is called to the long sand line. Frequently Stone is not 
available or is very expensive, when gravel is cheap and plentiful. By 
mixing gravel from two parts of the bank, the voids can often be reduced 
to 15% or 20^. With E equal to 10^ this makes splendid concrete, mixing 
in the cement directly, from 1 - 4-5 to 1 - 6, and, by stretching a 
point, even higher. 

It will be noted that the voids in sand do not run above A2% in the 
diagram. Frequently the first test of sand will show 45^ or SOJiS voids, 
but this can nearly always be reduced by mixing two kinds of sand, which 
is more economical than using the extra amount of cement involved. 

The number of cubic feet of cement paste which can be obtained from 
a barrel of cement depends upon a nximber of factors, the thoroughness of 
ramming, the voids in the cement itself, amount of water used, etc. The 
value given, 3.5 cu. ft., is one which best accords with the writer's ex- 
perience, and is, if anything, on the "safe side^" 

Classification of Concrete for Various Uses 

For purposes of estimation, the writer classifies concrete by the 
value of E, and in writing specifications and in the field uses the ordi- 
nary nomenclature of sand and stone, the values being taken from the dia- 
grams. 



(10) 



For fotindations 10^ concrete is reconiihended. 

For abutments and piers, 15/$ concrete is recommended. * 

For reinforced work, 20/? concrete is recommended. 

For thin sections, slabs and waterproof work, Z5% concrete is recom- 
mended, with an excess of water. 

Test for Voids 

The voids in the stone can be easily obtained by anyone in the field, 
but the sand voids are difficult to get, because with damp sand there are 
air bubbles within the mass that prevent the entrance of water into all the 
voids. A method that has been much recommended is to weigh a cubic foot of 
sand and figure out the voids by means of the specific gravity of quartz, 
2.65, or 165 lbs, per cu, ft. 

The best method known to the writer is as follows: Take a quart meas- 
ure exactly half full of water, and pour into it exactly one pint of the dry 
sand. If the water rises to the first gill mark from the top the voids are 
25^j if 1 1/2 gill marks from the top, voids are 1.5-^4^ 37.5?$, and so on. 

EXAMPLE : To illustrate the use of the diagrams. Suppose that we wish 
to make foundation concrete, and find the voids in the stone and sand 4:6% 
and Z5% respectively. Then, assuming E equals 10^, by Cr - 15 the propor- 
tions by volume are 1:2.75:5.75. 

For a check, the voids in 2.75 volumes of sand are 

2.75 X 0.33 equals 0.91 

E equals 10^ 0.09 

Amount of cement paste required 1.00, which 

checks* 

Also, the net volume of sand is 2.75 - 0.91 equals 1.84 vols. 
add cement paste 1.00 

Voids in stone plus 10^ E equals 2.84 

Therefore voids in stone equal 2. 84 "^1.1 equals 2o58 
Stone equals 2.58-^0.45 equals 5.75 vols, which 

checks with the diagram. 

We wish now to get the nxunber of barrels of cement per ou. yd. of 
concrete. On Cr - 17 for E equals 10^, corresponding to the 5,75 ordi- 
nate, we have the abscissa 1.28, or practically 1.3, Had we taken E 
equals 20^, with the same voids we should have had 1:2.5:5 concrete and 
1.40 bbls. per yard. 

The amounts of stone per cu. yd. of concrete are 0.96 and 0.92, 
found in brackets on the stone line of the diagrams. 

Take the data of the first example and suppose there is a gravel 
bank available not more than 1/4 mile from the work. We can get screened 
gravel delivered for, say 50 cts. per cu. yd. on the wagon. Mix 1/4 ou. 
yd. of this gravel having say 35^ voids, with every yard of the stone. 



(11) 



0.45 Cu, 


yd. 


0.55 " 




0.25 " 




0.088 " 




0.162 " 




1.05 " 




0.34 " 





Then in every cubic yard of stone there will be: 

Voids 

Net stone 

Gravel 

Voids in gravel ,25 x .35 

Net vol. gravel 

Gross vol, of stone and gravel, say 

Voids in stone and gravel, 1,05 - 0.71 

Voids = 0,34-M,05 = 32.4js, or say 35^. Our mixture with the same 
sand then becomes 1:2,75:7,4, instead of 1:2.75:5,75, and the cement is 
reduced from 1,3 bbls. to 1 bbl, per yard of finished concrete. 

The cost of the screened gravel has -been •97-r4 x 50 cts. or 12.1 
cts. The extra labor of mixing will on the most liberal estimate not be 
more than 10 cts. per yard. The saving in cement will be 0.3 bbl, per 
yard, or, if the cement cost fl,50 per bbl. on the work, 45 cts. - a net 
saving to somebody of 32.9 cts. per yard of concrete. On a thousand yard 
job this is a saving of $329, at an expenditure, on the part of the engi- 
neer, of an hour*s time and a little gray matter. On a ten thousand yard 
job the saving is no small item, and even on a small piece of work the 
saving pays the cost of the engineer's time many fold. Bear in mind that 
the mortar is just as rich as it was before, and that the resulting con- 
crete should be just as strong. These figures are finely drawn, but they 
are well within the limits of practice. 

The diagrams also give an idea of some of the pitfalls attending the 
use of our old standbys, the 1:2:5 and 1:3:6 mixtures, without first ascer- 
taining the voids. The 1:2:5 mixture contains an unnecessarily large 
amount of cement in the mortar and too much stone where the sand voids are 
34^ or less. The 1:3:6 mixture is all right for 30J^ sand and 47^ stone, 
but for Z5% stone 8 parts may be used, E being 10^, 



Cr - 18 shows the comparative strengths of different mixtures of con- 
crete as recommended by the Joint Committee of the A, S. C. E, The combi- 
nation of the voltime of fine and coarse aggregates measured separately for 
a given mixture of concrete is the abscissa and the limit of strength in 
potinds is shovm by the ordinates. At the right hand side of the chart are 
shown the values of n as recommended by the Joint Committee of the A. S. C, 
E, for different strengths of concrete. 



Cr - 19 is for determining the cost per ou, yd. of materials for con- 
crete of various mixes. 

I. By following steps indicated by arrows the voltmie of batch con- 
taining 1 bbl. of cement is obtained. 

II. By following steps indicated by arrows the cost of sand and 
aggregate per batch containing 1 bbl. of cement is obtained. 

III. From point indicating volume of batch found from I, Follow 
verti'^a? line to intersect horizontal line through the point giving cost 
of cement, sand and aggregate, (Value from II. plus cost of 1 bbl. of ce- 
ment.) From this intersection follow inclined line indicating Cost of 



Materials per cu, yd. to the vertical line "Cost of Cement, Sand and Aggre- 
gate per Cu. Yd." where the cost per cu. yd. is at once read. The cost per 
cu. yd. may also be read by interpolation of the values shown by the lines 
showing "Cost of Materials per Cu. Yd." between which the intersection of 
the vertical line (from the "Vol. of Batch Cohtaining 1 bbl. of Cement") and 
the horizontal line (From the "Cost of Cement, Sand and Agg. per Batch Con- 
taining 1 bbl.") falls, 

EXAMPLE: Wnat will be the cost of 1 cu. yd. of 1:3:6 concrete with the 
average of voids in sand and stone 40?S, cost of sand |2.25 per cu. yd., stone 
11.75 per cu. yd., cement $2.00 per bbl.? 

I. Enter the left hand margin of I at 9 = (3 + 6). Follovf horizontal 
line to right to intersection with inclined line indicating average voids of 
40^. From this intersection follow vertical line to bottom of diagram I 
where volume of batch is shown to be 0.9 cu. yd. 

II. Enter the right hand margin of II at 9 and follow horizontal line 
to left to intersection with inclined line |2.00, (the average price of sand 
and aggregate per cu. yd,). From this intersection follow vertical line to 
bottom of diagram II where the cost of sand and aggregate per batch contain- 
ing 1 bbl. of cement is shown to be $2,50, 

III, Enter top of diagram III at 0.9 and follow the vertical line dovn 
to the intersection with the horizontal line from a point representing the 
"Cost of Cement, Sand and Agg. per Batch Containing 1 bbl,", i.e,|4,50 
(^2.50-h|2.00). The cost per cu. yd. is found to be $5.00. 

FOaiULAS FOR REINFORCED CONCRETE CONSTRUCTION* 

1, STANDARD NOTATION 

(a) Rectangular Beams. 

The following notation is recommended: 

fg s tensile unit stress in steel; 

fj; a compressive iinit stress in concrete; 

Eg = modulus of elasticity of steel; 

Ec 9 modulus of elasticity of concrete; 

n - ^ J 

M = moment of resistance, or bending moment in general; 

As • steel area; 

b - breadth of beam; 

d = depth of beam to center of steel; 

k s ratio of depth of neutral axis to depth, d; 

2 = depth below top to resultant of the compressive stresses; 

3 s ratio of lever arm of resisting couple to depth, d; 
jd a d - z - arm of resisting couple; 

p = steel rati® • A_ • 

bd 

(b) T-Beams. 

b ■ width of flange; 

b ■ width of stem; 

t s thickness of flange. 

* Suggested in 1917 by Committee of A. S. C. E. 

(13) 



(c) Beams Reinforced for Compression, 

a' = area of compressive steel: 

p' ■ steel ratio for compressive steel: 

fs'« compressive unit stress in steel; 

C -z total compressive stress in concrete; 

C' 3 total compressive stress in steel; 

d' - depth to center of compressive steel; 

z s depth to resultant of C and C', 

(d) Shear^ Bond and Web Reinforcement. 

V • total shear; 

V' a total shear producing stress in reinforcement; 

V s shearing unit stress; ■ 

u s bond stress per unit area of bar; 

o a circmnferenoe of perimeter of bar; 

2o = sum of the perimeters of all bars; 

T s total stress in single reinforcing member; 

s s horizontal spacing of reinforcing members. 



(e) Columns. 



A s total net area; 

Ag a area of longitudinal steel; 

Ac s area of concrete; 

P s total safe load. 



<— — 


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• • • 


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2. FORMULAS, 
(a) Rectangular Beams, 




Position of neutral axis, k = y 2pn-f-(pn) - pn ..,,(1) 

Arm of resisting couple, j • 1 -2ik (2) 

3 
(For fg 3 15000 to 16000 and f© = 600 to 550, j may be taken at Z 7] 



8 



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(3) 



Fiber Stresses, fg s Agjd " pjbd2 

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jkbd? k 



f c = 



Steel ratio, for balanced reinforcement, p • 3: 1 

2 fs/fs -Ml 



,(5) 



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